Other than our overnight stay at Ninnaji Temple in Kyoto, most places will have a public computer in the lobby, either free or inexpensive to use.
Be warned: Japanese keyboards are WEIRD.
There’s often wireless or wired service in the lobby, if not in individual rooms.
Tokyo: New Izu Hotel had a computer in the lobby, along with wireless. Some of the rooms had wired service and we’re pretty sure they were extending that to all rooms gradually.
Hakone: we’re staying in a different set of buildings than we’re used to, so we don’t know exactly what’s available. There’s wireless in the “living room” of the building with the baths, though it was touchy to connect to.
Tajimi: You’ll be staying in private homes, so we can’t predict what will be available. Internet access in Japan is fairly cheap and fast by our standards, so you’ll probably have access.
Kyoto: We’re staying in a new (to us) hotel, but it should have at least lobby access, if not in each room.